Lightly oil a 12-by-9-inch oval baking dish. In the bottom of the baking dish, scatter the garlic halves. Starting at one side of the dish, create 3 lengthwise rows of vegetables that are standing on edge, alternating the slices of eggplant, onion and tomato until the dish is full. Brush the vegetables with 2 tablespoons of the oil and sprinkle with the salt. Bake for 45 to 55 minutes, or until the eggplant is very tender.

Carefully remove the baking dish from the oven and randomly insert the cheese slices into the vegetable rows. Brush with remaining 1 tablespoon oil and sprinkle with the thyme leaves. Return the baking dish to the oven with the thyme leaves. Return the baking dish to the oven and bake for 12 to 15 minutes, or until the cheese melts. Cool for at least 10 minutes before serving. Top with thyme springs, if desired.